#482202

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#36019	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18,22,27,23,26,37,42)(19,28,30,43,21,24,35,39)(20,31,32,44,33,25,36,41)(29,38,45,47,34,40,46,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,23,21)(18,24,26,28)(20,29,33,34)(22,30,37,35)(25,38,31,40)(27,43,42,39)(32,45,36,46)(41,48,44,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,33)(18,25,26,31)(19,29,21,34)(22,36,37,32)(24,38,28,40)(27,44,42,41)(30,46,35,45)(39,48,43,47);(17,21,23,19)(18,24,26,28)(20,29,33,34)(22,35,37,30)(25,40,31,38)(27,39,42,43)(32,45,36,46)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389398);